Unit 3: Lesson 1Proportional Relationships

Learning Targets ….

• Students work with proportional relationships that involve average speed and constant speed in order to write a linear equation in two variables.

• Students use linear equations in two variables to answer questions about distance and time.

HousekeepingYesterday, we introduced Unit 3.

Next week - No Live Classes!

Goal for today……Agenda Review

Ratio vs. Rate Unit Rate Proportions

Proportional Relationships

DiscussionPaul walks 2 miles in 25 minutes. How many miles can Paul walk in 137.5 minutes?

How many miles would he walk in 50 minutes?

How many miles would he walk in 75 minutes?

How many miles would he walk in 100 minutes?

How many miles would he walk in 125 minutes?

DiscussionPaul walks 2 miles in 25 minutes. How many miles can Paul walk in 137.5 minutes?

Time (in minutes) Distance (in miles)

A ratio is a comparison between two similar things.

What is a Ratio?

Ratio: 4 3

Rate: 90 miles3 hours

Read as “90 miles per 3 hours.”

A rate is a comparison of two quantities that have different units.

Since the relationship between the distance Paul walks and the time it takes him to walk that distance is proportional, we let y represent the distance Paul walks in 137.5 minutes and write the following:

Proportional Relationship

252 =

137.5𝑦

Proportions

Identify Proportions

Tell whether the ratios are proportional.

410

615

Since the cross products are equal, the ratios are proportional.

60

=?

Example: Using Cross Products to Identify Proportions

60 = 60

Find cross products.604

10615

Identify ProportionsExample: Using Cross Products to Identify Proportions

A mixture of fuel for a certain small engine should be 4 parts gasoline to 1 part oil. If you combine 5 quarts of oil with 15 quarts of gasoline, will the mixture be correct?4 parts gasoline

1 part oil =? 15 quarts gasoline5 quarts oil

4 • 5 = 20 1 • 15 = 15

20 ≠ 15The cross products are not equal. The mixture will not be correct.

Set up equal ratios.

Find the cross products.

Solving a ProportionThe ratio of boys to girls in a soccer league is 5:4. If there are 52 boys in the league, how many girls are there?

Write a ratio comparing girls to boys.

Set Up the proportion. Let x represent the number of girls.

54

girlsboys =

Since x is divided by 52, multiply both sides of the equation by 56.

54 = x

52

65 = x

(52) (52)54 = x

52

There are 65 girls in the league.

Solving a ProportionA cable car travels 15 miles in 80 minutes. At this rate of speed, how long would it take the cable car to travel 45 miles?

15 ▪ s = 80 ▪ 45

Set up a proportion that compares distance to time.

Let s represent the time the takes to travel 45 mi.

distance 1time 1 = distance 2

time 2

At a constant rate of speed ratios of distance to time are equivalent.

Find the cross products.

15 mi80 min= 45 mi

s

How many miles, y, can Paul walk in x minutes?

Think about it?

We know for a fact that Paul can walk 2 miles in 25 minutes, so we can write the ratio as we did with the proportion. We can write another ratio for the number of miles, y, Paul walks in x minutes. It is . For the same reason we could write the proportion before, we can write one now with these two ratios:

How many miles, y, can Paul walk in x minutes?

Think about it?

Does this remind you of something we have recently done?

It’s a linear equation in disguise!!!!!

Paul can walk y miles in 0.08x minutes. This equation will allow us to answer all kinds of questions about Paul with respect to any given number of minutes or miles.

Time (in minutes) Distance (in miles)

25 2

50 4

75 6

100 8

125 10

Let’s go back to the table and look for y = 0.08x or its equivalent y = x. What do you notice?

The fraction came from the first row in the table. It is the distance traveled divided by the time it took to travel that distance. It is also in between each row of the table.

What is a Unit Ratio?Unit rates are rates in which the second quantity is 1.

unit rate: 30 miles,1 hour or 30 mi/h

The ratio 903 can be simplified by dividing:

903 = 30

1

Finding a Unit RatePenelope can type 90 words in 2 minutes. How many words can she type in 1 minute?

90 words 2 minutes Write a rate.

=

Penelope can type 45 words in one minute.

90 words ÷ 2 2 minutes ÷ 2

Divide to find words per minute.

45 words 1 minute

Finding a Unit RateGeoff can type 30 words in half a minute. How many words can he type in 1 minute?

Write a rate.

=

Geoff can type 60 words in one minute.

Multiply to find words per minute.

60 words 1 minute

30 words minute 12

30 words • 2 minute • 212

Check out this tableTime (in hours) Distance (in miles)

3 123

6 246

9 369

12 492

y

We want to know how many miles, y, can be traveled in any number of hours x.

Set it up

What does the equation y = 41x mean?

ExampleConsider the following word problem: Alexxa walked from Grand Central Station on 42nd Street to Penn Station on 7th Avenue. The total distance traveled was 1.1 miles. It took Alexxa 25 minutes to make the walk. How many miles did she walk in the first 10 minutes?

ExampleAre you sure about your answer? How often do you walk at a constant speed? Notice the problem did not even mention that she was walking at the same rate throughout the entire 1.1 miles.

What if you have more information about her walk: Alexxa walked from Grand Central Station (GCS) along 42nd Street to an ATM machine 0.3 miles away in 8 minutes. It took her 2 minutes to get some money out of the machine. Do you think your answer is still correct?

ExampleLet’s continue with Alexxa’s walk: She reached the 7th Avenue junction 13 minutes after she left GCS, a distance of 0.6 miles. There, she met her friend Karen with whom she talked for 2 minutes. After leaving her friend, she finally got to Penn Station 25 minutes after her walk began.

Is this a more realistic situation than believing that she walked the exact same speed throughout the entire trip? What other events typically occur during walks in the city?

Example

Time (in minutes) Distance Traveled (in miles)

0 0

8 0.3

10 0.3

13 0.6

15 0.6

25 1.1

The following table shows an accurate picture of Alexxa’s walk:

Average SpeedNow that we have an idea of what could go wrong when we assume a person walks at a constant rate or that a proportion can give us the correct answer all of the time, let’s define what is called average speed.

Suppose a person walks a distance of d (miles) in a given time interval t (minutes). Then, the average speed in the given time interval is in miles per minute.

ExampleSuppose a person walks a distance of d (miles) in a given time interval t (minutes). Then, the average speed in the given time interval is in miles per minute.

With this definition we can calculate Alexxa’s average speed: The distance that Alexxa traveled divided by the time interval she walked is miles per minute.

ExampleIf we assume that someone can actually walk at the same average speed over any time interval, then we say that the person is walking at a constant speed. Suppose the average speed of a person is the same constant C for any given time interval. Then, we say that the person is walking at a constant speed C.

ExampleIf the original problem included information specifying constant speed, then we could write the following: Alexxa’s average speed for 25 minutes is . Let y represent the distance Alexxa walked in 10 minutes. Then, her average speed for 10 minutes is . Since Alexxa is walking at a constant speed of C miles per minute, then we know that

= C, and = C.

ExampleSince both fractions are equal to C, then we can write

= .

With the assumption of constant speed, we now have a proportional relationship, which would make the answer you came up with in the beginning correct.

ExampleWe can go one step further and write a statement in general. If Alexxa walks y miles in x minutes, then

and = .

To find how many miles Alexxa walks in x miles, we 𝑦𝑦solve the equation for y:

Wrap Up• Average speed is found by taking the total distance

traveled in a given time interval, divided by the time interval.

• If we assume the same average speed over any time interval, then we have constant speed, which can then be used to express a linear equation in two variables relating distance and time.

• We know how to use linear equations to answer questions about distance and time.

• We cannot assume that a problem can be solved using a proportion unless we know that the situation involves constant speed (or rate).

What now?Questions???

We will work on this lesson for two days

Day 1:Need to practice some basic ratio and proportion skills??? ** IXL Skills: H.1 – H.9

Day 2:Complete the Proportional Relationships Practice Problems